A Mathematical Theory of Ranking

Yin Cheng

Home/Archive/arXiv

arXiv · 2604.09733

A Mathematical Theory of Ranking

Yin Cheng

View original source ↗Read source document ↗Export metadata ↓

Abstract

Ranking systems produce ordered lists from scalar scores, yet the ranking itself depends only on pairwise comparisons. We develop a mathematical theory that takes this observation seriously, centering the analysis on pairwise margins rather than absolute scores. In the linear case, each pairwise margin decomposes exactly into factor-level contributions. We prove that the resulting L_1 local influence share is the unique budgeting rule consistent with pure factor refinement. Aggregating local shares yields a global influence structure: in log-absolute-weight coordinates, this structure is the gradient of a convex potential, its Jacobian is a competition-graph Laplacian, and Influence Exchange -- the reallocation of pairwise control across model states -- satisfies a finite energy identity with a zero-exchange rigidity law. For nonlinear scoring, the pairwise margin remains well-defined, but factor-level decomposition becomes path-dependent due to cross-factor interactions. We prove an interaction-curvature theorem: factorwise path attribution is path-independent if and only if the relevant mixed partial derivatives vanish, recovering full factorwise uniqueness exactly in the additive regime. The framework extends through local linearization and Pairwise Integrated Gradients. The geometric arc continues through permutation space, score-space hyperplane crossings, discrete exactness and triangle curl, Hodge-like diagnostics, and root-space/Weyl-chamber geometry -- organized as successive interpretive closures of the same pairwise-first analytical progression.

Explore related subjects

cs.IR

Keep this discovery

BibTeX RIS

Yin Cheng. 2026-04-09. A Mathematical Theory of Ranking. https://arxiv.org/abs/2604.09733

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

UniRec: Cross-stage Multi-Task Fusion with Preference Alignment for Cascaded Recommender Systems

Industrial recommender systems use cascaded stages with different objectives, feature spaces, and latency constraints. Optimizing pre-ranking and ranking separately can create cross-stage inconsistency: upstream models may filter out items preferred by downstream rankers, and independently tuned downstream fusion can offset upstream improvements. Existing multi-task fusion methods focus on multi-objective fusion within the ranking stage, and cross-stage methods typically only add a downstream score factor to upstream ranking. Joint optimization of fusion modules across both stages remains largely unexplored. We propose UniRec, a Unified Cross-stage Recommendation Fusion model. First, the two fusion agents partially share input embeddings and are trained in a single computation graph, so gradients from either stage propagate through the shared representation and influence the other. Second, we introduce a dual-axis preference alignment objective: a vertical cross-stage consistency term transfers downstream pairwise preferences to the upstream fusion score, and a horizontal compact aggregation term reorganizes dozens of pairwise objectives over heterogeneous prior signals into bidirectional preference evidence. Third, we find that unconstrained end-to-end fusion optimization can exploit imbalances in item attribute distributions, over-concentrating on high-reward regions at the cost of other objectives. We therefore add an attribute group-relative regularization that computes advantages within attribute groups and normalizes the policy over the same groups, so uniformly promoting an entire high-reward group yields no optimization gain. Offline, UniRec consistently outperforms single-stage fusion and cross-stage coordination baselines. Online A/B tests show a 0.616\% gain in app usage duration. UniRec is fully deployed on the Kuaishou platform.

cs.IR↗

arXiv2026

VikingRAG: Accurate and Token-efficient Retrieval-augmented Generation over Structured Documents

State-of-the-art retrieval-augmented generation (RAG) methods exploit document structures to acquire sufficient evidence, but often incur substantial token costs. To reduce structural-context tokens without compromising high RAG accuracy, we present {\sf VikingRAG}, a directory-aware semantic data management system that tightly integrates semantic and structural access to support structural-context-efficient, evidence-gap-driven multi-round retrieval. To further reduce token overhead of multi-round interaction, we materialize agentic multi-round retrieval traces as experience edges, and reuse these edges for similar queries, avoiding repeated multi-round exploration. To additionally reduce token costs when agentic multi-round retrieval is unnecessary, we introduce an adaptive escalation strategy that answers from one-round experience-augmented retrieval when the evidence is sufficient, and invokes agentic multi-round retrieval only otherwise. Experiments on real datasets show that the base system {\sf VikingRAG} matches high accuracy of state-of-the-art methods while consuming only 11.6\%--51.9\% of their tokens. With retrieval-trace reuse and adaptive escalation, token costs drop to 5.1\%--32.5\% while maintaining competitive accuracy and practical document-storage performance, showing the utility of this work for emerging AI knowledge bases.

cs.IR↗

arXiv2026

TimelyRAG: Semantic-Temporal Hybrid Retrieval for Time-Critical Question Answering in Overlapping-Evolving Documents

Although large language models (LLMs) and retrieval-augmented generation (RAG) have advanced open-domain question answering (QA), they remain unreliable when documents evolve through amendments. Existing time-sensitive retrieval methods address only the disjoint-evolving environment, where each update is an independent snapshot. However, laws, policies, and regulations often operate in overlapping-evolving environments, where amendments override earlier clauses while preserving most content, creating strong semantic overlap across versions. We propose TimelyRAG, a retriever-agnostic framework that incorporates temporal distance into ranking to align queries with version-appropriate documents. We also introduce TimelyQABench, the first benchmark for regulation-heavy domains with overlapping-evolving challenges. Experiments show consistent gains, up to +28.6% in nDCG@10, highlighting the importance of temporal reasoning for reliable QA over evolving documents. All resources are available at https://github.com/kaist-dmlab/TimelyRAG.

cs.IR↗